A fully consistent interpretation of the Dirac equation is only possible in the framework of quantum field theory. This is due to the possibility of negative-energy solutions, or, in mathematical terms, to the fact that the Dirac operator is not bounded from below. Far from the pair-production threshold, however, one can recover an approximate single-particle picture for the relativistic electron by restricting the Dirac Hamiltonian to its positive spectral subspace. Alternatively, one can employ a Foldy-Wouthuysen-type transformation to derive an approximate operator in the positive spectral subspace of the free Hamiltonian, which contains the effects of the Coulomb field up to the second order [G. Jansen and B. Hess, Phys. Rev. A 39 (1989), no. 11, 6016--6017]. The positivity of the Jansen-Hess (JH) operator has been proved recently for nuclear charges up to the critical value $Z_c = 25$Zc=25 [R. Brummelhuis, H. K. H. Siedentop and E. Stockmeyer, Doc. Math. 7 (2002), 167--182 (electronic); MR1911215 (2004b:81046)]. In this paper, the proof of positivity is extended up to $Z_c = 114$Zc=114 by considering the JH Hamiltonian in a momentum-space, partial wave representation. One first proves positivity of the kernels of the partial-wave components of the JH operator. As a second step, one then establishes the monotonicity properties of these kernels with respect to the orbital quantum number. This is enough to prove positivity analytically for $Z\leq 81$Z≤81, and through the numerical solution of a transcendental equation for $Z \leq 114$Z≤114.